Fearless symmetry: exposing the hidden patterns of numbers
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Princeton [u.a.]
Princeton Univ. Press
2006
|
| Subjects: | |
| Online Access: | Contributor biographical information Publisher description Table of contents Inhaltsverzeichnis |
| Physical Description: | XXV, 272 S. graph. Darst. |
| ISBN: | 0691124922 9780691124926 |
Staff View
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Record in the Search Index
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| adam_text | S 8AHHEXKA PATTERNS OF NUMBERS ROBERT GROSS PRINCETON AND OXFORD
CONTENTS FOREWORD BY BARRY MAZUR XV PREFACE XXI ACKNOWLEDGMENTS XXVII
GREEK ALPHABET XXIX PART ONE. ALGEBRAIC PRELIMINARIES CHAPTER 1.
REPRESENTATIONS 3 THE BARE NOTION OF REPRESENTATION 3 AN EXAMPLE:
COUNTING 5 DIGRESSION: DEFINITIONS 6 COUNTING (CONTINUED) 7 COUNTING
VIEWED AS A REPRESENTATION 8 THE DEFINITION OF A REPRESENTATION 9
COUNTING AND INEQUALITIES AS REPRESENTATIONS 10 SUMMARY 11 CHAPTER 2.
GROUPS 13 THE GROUP OF ROTATIONS OF A SPHERE 14 THE GENERAL CONCEPT OF
GROUP 17 IN PRAISE OF MATHEMATICAL IDEALIZATION 18 DIGRESSION: LIE
GROUPS 19 CONTENTS CHAPTER 3. PERMUTATIONS THE ABC OF PERMUTATIONS
PERMUTATIONS IN GENERAL CYCLES DIGRESSION: MATHEMATICS AND SOCIETY
CHAPTER 4. MODULAR ARITHMETIC CYCLICAL TIME CONGRUENCES ARITHMETIC
MODULO A PRIME MODULAR ARITHMETIC AND GROUP THEORY MODULAR ARITHMETIC
AND SOLUTIONS OF EQUATIONS CHAPTER 5. COMPLEX NUMBERS OVERTURE TO
COMPLEX NUMBERS COMPLEX ARITHMETIC COMPLEX NUMBERS AND SOLVING EQUATIONS
DIGRESSION: THEOREM ALGEBRAIC CLOSURE CHAPTER 6. EQUATIONS AND VARIETIES
THE LOGIC OF EQUALITY THE HISTORY OF EQUATIONS Z-EQUATIONS VARIETIES
SYSTEMS OF EQUATIONS EQUIVALENT DESCRIPTIONS OF THE SAME VARIETY FINDING
RV OTS OF POLYNOMIALS ARE THERE GENERAL METHODS FOR FINDING SOLUTIONS TO
SYSTEMS OF POLYNOMIAL EQUATIONS? DEEPER UNDERSTANDING IS DESIRABLE
CHAPTER 7. QUADRATIC RECIPROCITY THE SIMPLEST POLYNOMIAL EQUATIONS WHEN
IS * 1 A SQUARE MOD P? THE LEGENDRE SYMBOL DIGRESSION: NOTATION GUIDES
THINKING MULTIPLICATIVITY OF THE LEGENDRE SYMBOL 21 21 25 26 29 31 31 33
36 39 41 42 42 44 47 47 47 49 50 50 52 54 56 58 61 62 65 67 67 69 71 72
73 CONTENTS WHEN IS 2 A SQUARE MOD PI 74 WHEN IS 3 A SQUARE MOD PI 75
WHEN IS 5 A SQUARE MOD PI (WILL THIS GO ON FOREVER?) 76 THE LAW OF
QUADRATIC RECIPROCITY 78 EXAMPLES OF QUADRATIC RECIPROCITY 80 PART TWO.
GALOIS THEORY AND REPRESENTATIONS CHAPTER 8. GALOIS THEORY 87
POLYNOMIALS AND THEIR ROOTS 88 THE FIELD OF ALGEBRAIC NUMBERS Q ALG 89
THE ABSOLUTE GALOIS GROUP OF Q DEFINED 92 A CONVERSATION WITH S: A
PLAYLET IN THREE SHORT SCENES 93 DIGRESSION: SYMMETRY 96 HOW ELEMENTS OF
G BEHAVE 96 WHY IS G A GROUP? 101 SUMMARY 101 CHAPTER 9. ELLIPTIC CURVES
103 ELLIPTIC CURVES ARE GROUP VARIETIES 103 AN EXAMPLE 104 THE GROUP
LAW ON AN ELLIPTIC CURVE 107 A MUCH-NEEDED EXAMPLE 108 DIGRESSION: WHAT
IS SO GREAT ABOUT ELLIPT C CURVES? 109 THE CONGRUENT NUMBER PROBLEM 110
TORSION AND THE GALOIS GROUP 111 CHAPTER 10. MATRICES 114 MATRICES AND
MATRIX REPRESENTATIONS 114 MATRICES AND THEIR ENTRIES 115 MATRIX
MULTIPLICATION 117 LINEAR ALGEBRA 120 DIGRESSION: GRAECO-LATIN SQUARES
122 CHAPTER 11. GROUPS OF MATRICES 124 SQUARE MATRICES 124 MATRIX
INVERSES 126 CONTENTS THE GENERAL LINEAR GROUP OF INVERTIBLE MATRICES
129 THE GROUP GL(2,Z) 130 SOLVING MATRIX EQUATIONS 132 CHAPTER 12. GROUP
REPRESENTATIONS 135 MORPHISMS OF GROUPS 135 A 4 , SYMMETRIES OF A
TETRAHEDRON 139 REPRESENTATIONS OF A 4 142 MODP LINEAR REPRESENTATIONS
OF THE ABSOLUTE GALOIS GROUP FROM ELLIPTIC CURVES 146 CHAPTER 13. THE
GALOIS GROUP OF A POLYNOMIAL 149 THE FIELD GENERATED BY A Z-POLYNOMIAL
149 EXAMPLES 151 DIGRESSION: THE INVERSE GALOIS PROBLEM 154 TWO MORE
THINGS 155 CHAPTER 14. THE RESTRICTION MORPHISM 157 THE BIG PICTURE AND
THE LITTLE PICTURES 157 BASIC FACTS ABOUT THE RESTRICTION MORPHISM 159
EXAMPLES 161 CHAPTER 15. THE GREEKS HAD A NAME FOR IT 162 TRACES 163
CONJUGACY CLASSES 165 EXAMPLES OF CHARACTERS 166 HOW THE CHARACTER OF A
REPRESENTATION DETERMINES THE REPRESENTATION 171 PRELUDE TO THE NEXT
CHAPTER 175 DIGRESSION: A FACT ABOUT ROTATIONS OF THE SPHERE 175 CHAPTER
16. FROBENIUS 177 SOMETHING FOR NOTHING 177 GOOD PRIME, BAD PRIME 179
ALGEBRAIC INTEGERS, DISCRIMINANTS, AND NORMS 180 A WORKING DEFINITION OF
FROB P 184 AN EXAMPLE OF COMPUTING FROBENIUS ELEMENTS 185 FROB P AND
FACTORING POLYNOMIALS MODULO P 186 CONTENTS APPENDIX: THE OFFICIAL
DEFINITION OF THE BAD PRIMES FOR A GALOIS REPRESENTATION 188 APPENDIX:
THE OFFICIAL DEFINITION OF UNRAMIFIED AND FROB P 189 PART THREE.
RECIPROCITY LAWS CHAPTER 17. RECIPROCITY LAWS 193 THE LIST OF TRACES OF
FROBENIUS 193 BLACK BOXES 195 WEAK AND STRONG RECIPROCITY LAWS 196
DIGRESSION: CONJECTURE 197 KINDS OF BLACK BOXES 199 CHAPTER 18. ONE- AND
TWO-DIMENSIONAL REPRESENTATIONS 200 ROOTS OF UNITY 200 HOW FROBQ ACTS ON
ROOTS OF UNITY 202 ONE-DIMENSIONAL GALOIS REPRESENTATIONS 204
TWO-DIMENSIONAL GALOIS REPRESENTATIONS ARISING FROM THEP-TORSION POINTS
OF AN ELLIPTIC CURVE 205 HOW FROB Q ACTS ONP-TORSION POINTS 207 THE
2-TORSION 209 AN EXAMPLE 209 ANOTHER EXAMPLE 211 YET ANOTHER EXAMPLE 212
THE PROOF 214 CHAPTER 19. QUADRATIC RECIPROCITY REVISITED 216
SIMULTANEOUS EIGENELEMENTS 217 THE Z-VARIETY X 2 - W 218 A WEAK
RECIPROCITY LAW 220 A STRONG RECIPROCITY LAW 221 A DERIVATION OF
QUADRATIC RECIPROCITY 222 CHAPTER 20. A MACHINE FOR MAKING GALOIS
REPRESENTATIONS 225 VECTOR SPACES AND LINEAR ACTIONS OF GROUPS 225
CONTENTS LINEARIZATION 228 ETALE COHOMOLOGY 229 CONJECTURES ABOUT ETALE
COHOMOLOGY 231 CHAPTER 21. A LAST LOOK AT RECIPROCITY 233 WHAT IS
MATHEMATICS? 233 RECIPROCITY 235 MODULAR FORMS 236 REVIEW OF RECIPROCITY
LAWS 239 A PHYSICAL ANALOGY 240 CHAPTER 22. FERMAT S LAST THEOREM AND
GENERALIZED FERMAT EQUATIONS 242 THE THREE PIECES OF THE PROOF 243 FREY
CURVES 244 THE MODULARITY CONJECTURE 245 LOWERING THE LEVEL 247 PROOF OF
FLT GIVEN THE TRUTH OF THE MODULARITY CONJECTURE FOR CERTAIN ELLIPTIC
CURVES 249 BRING ON THE RECIPROCITY LAWS 250 WHAT WILES AND TAYLOR-WILES
DID 252 GENERALIZED FERMAT EQUATIONS 254 WHAT HENRI DARMON AND LOIC
MEREL DID 255 PROSPECTS FOR SOLVING THE GENERALIZED FERMAT EQUATIONS 256
CHAPTER 23. RETROSPECT 257 TOPICS COVERED 257 BACK TO SOLVING EQUATIONS
258 DIGRESSION: WHY DO MATH? 260 THE CONGRUENT NUMBER PROBLEM 261
PEERING PAST THE FRONTIER 263 BIBLIOGRAPHY INDEX 265 269
|
| adam_txt |
S 8AHHEXKA PATTERNS OF NUMBERS ROBERT GROSS PRINCETON AND OXFORD
CONTENTS FOREWORD BY BARRY MAZUR XV PREFACE XXI ACKNOWLEDGMENTS XXVII
GREEK ALPHABET XXIX PART ONE. ALGEBRAIC PRELIMINARIES CHAPTER 1.
REPRESENTATIONS 3 THE BARE NOTION OF REPRESENTATION 3 AN EXAMPLE:
COUNTING 5 DIGRESSION: DEFINITIONS 6 COUNTING (CONTINUED) 7 COUNTING
VIEWED AS A REPRESENTATION 8 THE DEFINITION OF A REPRESENTATION 9
COUNTING AND INEQUALITIES AS REPRESENTATIONS 10 SUMMARY 11 CHAPTER 2.
GROUPS 13 THE GROUP OF ROTATIONS OF A SPHERE 14 THE GENERAL CONCEPT OF
"GROUP" 17 IN PRAISE OF MATHEMATICAL IDEALIZATION 18 DIGRESSION: LIE
GROUPS 19 CONTENTS CHAPTER 3. PERMUTATIONS THE ABC OF PERMUTATIONS
PERMUTATIONS IN GENERAL CYCLES DIGRESSION: MATHEMATICS AND SOCIETY
CHAPTER 4. MODULAR ARITHMETIC CYCLICAL TIME CONGRUENCES ARITHMETIC
MODULO A PRIME MODULAR ARITHMETIC AND GROUP THEORY MODULAR ARITHMETIC
AND SOLUTIONS OF EQUATIONS CHAPTER 5. COMPLEX NUMBERS OVERTURE TO
COMPLEX NUMBERS COMPLEX ARITHMETIC COMPLEX NUMBERS AND SOLVING EQUATIONS
DIGRESSION: THEOREM ALGEBRAIC CLOSURE CHAPTER 6. EQUATIONS AND VARIETIES
THE LOGIC OF EQUALITY THE HISTORY OF EQUATIONS Z-EQUATIONS VARIETIES
SYSTEMS OF EQUATIONS EQUIVALENT DESCRIPTIONS OF THE SAME VARIETY FINDING
RV OTS OF POLYNOMIALS ARE THERE GENERAL METHODS FOR FINDING SOLUTIONS TO
SYSTEMS OF POLYNOMIAL EQUATIONS? DEEPER UNDERSTANDING IS DESIRABLE
CHAPTER 7. QUADRATIC RECIPROCITY THE SIMPLEST POLYNOMIAL EQUATIONS WHEN
IS * 1 A SQUARE MOD P? THE LEGENDRE SYMBOL DIGRESSION: NOTATION GUIDES
THINKING MULTIPLICATIVITY OF THE LEGENDRE SYMBOL 21 21 25 26 29 31 31 33
36 39 41 42 42 44 47 47 47 49 50 50 52 54 56 58 61 62 65 67 67 69 71 72
73 CONTENTS WHEN IS 2 A SQUARE MOD PI 74 WHEN IS 3 A SQUARE MOD PI 75
WHEN IS 5 A SQUARE MOD PI (WILL THIS GO ON FOREVER?) 76 THE LAW OF
QUADRATIC RECIPROCITY 78 EXAMPLES OF QUADRATIC RECIPROCITY 80 PART TWO.
GALOIS THEORY AND REPRESENTATIONS CHAPTER 8. GALOIS THEORY 87
POLYNOMIALS AND THEIR ROOTS 88 THE FIELD OF ALGEBRAIC NUMBERS Q ALG 89
THE ABSOLUTE GALOIS GROUP OF Q DEFINED 92 A CONVERSATION WITH S: A
PLAYLET IN THREE SHORT SCENES 93 DIGRESSION: SYMMETRY 96 HOW ELEMENTS OF
G BEHAVE 96 WHY IS G A GROUP? 101 SUMMARY 101 CHAPTER 9. ELLIPTIC CURVES
103 ELLIPTIC CURVES ARE "GROUP VARIETIES" 103 AN EXAMPLE 104 THE GROUP
LAW ON AN ELLIPTIC CURVE 107 A MUCH-NEEDED EXAMPLE 108 DIGRESSION: WHAT
IS SO GREAT ABOUT ELLIPT'C CURVES? 109 THE CONGRUENT NUMBER PROBLEM 110
TORSION AND THE GALOIS GROUP 111 CHAPTER 10. MATRICES 114 MATRICES AND
MATRIX REPRESENTATIONS 114 MATRICES AND THEIR ENTRIES 115 MATRIX
MULTIPLICATION 117 LINEAR ALGEBRA 120 DIGRESSION: GRAECO-LATIN SQUARES
122 CHAPTER 11. GROUPS OF MATRICES 124 SQUARE MATRICES 124 MATRIX
INVERSES 126 CONTENTS THE GENERAL LINEAR GROUP OF INVERTIBLE MATRICES
129 THE GROUP GL(2,Z) 130 SOLVING MATRIX EQUATIONS 132 CHAPTER 12. GROUP
REPRESENTATIONS 135 MORPHISMS OF GROUPS 135 A 4 , SYMMETRIES OF A
TETRAHEDRON 139 REPRESENTATIONS OF A 4 142 MODP LINEAR REPRESENTATIONS
OF THE ABSOLUTE GALOIS GROUP FROM ELLIPTIC CURVES 146 CHAPTER 13. THE
GALOIS GROUP OF A POLYNOMIAL 149 THE FIELD GENERATED BY A Z-POLYNOMIAL
149 EXAMPLES 151 DIGRESSION: THE INVERSE GALOIS PROBLEM 154 TWO MORE
THINGS 155 CHAPTER 14. THE RESTRICTION MORPHISM 157 THE BIG PICTURE AND
THE LITTLE PICTURES 157 BASIC FACTS ABOUT THE RESTRICTION MORPHISM 159
EXAMPLES 161 CHAPTER 15. THE GREEKS HAD A NAME FOR IT 162 TRACES 163
CONJUGACY CLASSES 165 EXAMPLES OF CHARACTERS 166 HOW THE CHARACTER OF A
REPRESENTATION DETERMINES THE REPRESENTATION 171 PRELUDE TO THE NEXT
CHAPTER 175 DIGRESSION: A FACT ABOUT ROTATIONS OF THE SPHERE 175 CHAPTER
16. FROBENIUS 177 SOMETHING FOR NOTHING 177 GOOD PRIME, BAD PRIME 179
ALGEBRAIC INTEGERS, DISCRIMINANTS, AND NORMS 180 A WORKING DEFINITION OF
FROB P 184 AN EXAMPLE OF COMPUTING FROBENIUS ELEMENTS 185 FROB P AND
FACTORING POLYNOMIALS MODULO P 186 CONTENTS APPENDIX: THE OFFICIAL
DEFINITION OF THE BAD PRIMES FOR A GALOIS REPRESENTATION 188 APPENDIX:
THE OFFICIAL DEFINITION OF "UNRAMIFIED" AND FROB P 189 PART THREE.
RECIPROCITY LAWS CHAPTER 17. RECIPROCITY LAWS 193 THE LIST OF TRACES OF
FROBENIUS 193 BLACK BOXES 195 WEAK AND STRONG RECIPROCITY LAWS 196
DIGRESSION: CONJECTURE 197 KINDS OF BLACK BOXES 199 CHAPTER 18. ONE- AND
TWO-DIMENSIONAL REPRESENTATIONS 200 ROOTS OF UNITY 200 HOW FROBQ ACTS ON
ROOTS OF UNITY 202 ONE-DIMENSIONAL GALOIS REPRESENTATIONS 204
TWO-DIMENSIONAL GALOIS REPRESENTATIONS ARISING FROM THEP-TORSION POINTS
OF AN ELLIPTIC CURVE 205 HOW FROB Q ACTS ONP-TORSION POINTS 207 THE
2-TORSION 209 AN EXAMPLE 209 ANOTHER EXAMPLE 211 YET ANOTHER EXAMPLE 212
THE PROOF 214 CHAPTER 19. QUADRATIC RECIPROCITY REVISITED 216
SIMULTANEOUS EIGENELEMENTS 217 THE Z-VARIETY X 2 - W 218 A WEAK
RECIPROCITY LAW 220 A STRONG RECIPROCITY LAW 221 A DERIVATION OF
QUADRATIC RECIPROCITY 222 CHAPTER 20. A MACHINE FOR MAKING GALOIS
REPRESENTATIONS 225 VECTOR SPACES AND LINEAR ACTIONS OF GROUPS 225
CONTENTS LINEARIZATION 228 ETALE COHOMOLOGY 229 CONJECTURES ABOUT ETALE
COHOMOLOGY 231 CHAPTER 21. A LAST LOOK AT RECIPROCITY 233 WHAT IS
MATHEMATICS? 233 RECIPROCITY 235 MODULAR FORMS 236 REVIEW OF RECIPROCITY
LAWS 239 A PHYSICAL ANALOGY 240 CHAPTER 22. FERMAT'S LAST THEOREM AND
GENERALIZED FERMAT EQUATIONS 242 THE THREE PIECES OF THE PROOF 243 FREY
CURVES 244 THE MODULARITY CONJECTURE 245 LOWERING THE LEVEL 247 PROOF OF
FLT GIVEN THE TRUTH OF THE MODULARITY CONJECTURE FOR CERTAIN ELLIPTIC
CURVES 249 BRING ON THE RECIPROCITY LAWS 250 WHAT WILES AND TAYLOR-WILES
DID 252 GENERALIZED FERMAT EQUATIONS 254 WHAT HENRI DARMON AND LOIC
MEREL DID 255 PROSPECTS FOR SOLVING THE GENERALIZED FERMAT EQUATIONS 256
CHAPTER 23. RETROSPECT 257 TOPICS COVERED 257 BACK TO SOLVING EQUATIONS
258 DIGRESSION: WHY DO MATH? 260 THE CONGRUENT NUMBER PROBLEM 261
PEERING PAST THE FRONTIER 263 BIBLIOGRAPHY INDEX 265 269 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 180 |
| ctrlnum | (OCoLC)60826707 (DE-599)DNB 2005051471 |
| dewey-full | 512.7 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
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| discipline | Mathematik |
| discipline_str_mv | Mathematik |
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| id | DE-604.BV022542563 |
| illustrated | Illustrated |
| index_date | 2024-07-02T18:10:46Z |
| indexdate | 2024-07-09T20:59:52Z |
| institution | BVB |
| isbn | 0691124922 9780691124926 |
| language | English |
| lccn | 2005051471 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015748979 |
| oclc_num | 60826707 |
| open_access_boolean | |
| owner | DE-703 DE-188 |
| owner_facet | DE-703 DE-188 |
| physical | XXV, 272 S. graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Princeton Univ. Press |
| record_format | marc |
| spelling | Ash, Avner Verfasser aut Fearless symmetry exposing the hidden patterns of numbers Avner Ash and Robert Gross Princeton [u.a.] Princeton Univ. Press 2006 XXV, 272 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Nombres, Théorie des Number theory Symmetrie (DE-588)4058724-1 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Darstellungstheorie (DE-588)4148816-7 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 s Darstellungstheorie (DE-588)4148816-7 s Symmetrie (DE-588)4058724-1 s 1\p DE-604 Gross, Robert Verfasser aut http://www.loc.gov/catdir/enhancements/fy0654/2005051471-b.html Contributor biographical information http://www.loc.gov/catdir/enhancements/fy0654/2005051471-d.html Publisher description http://www.loc.gov/catdir/enhancements/fy0654/2005051471-t.html Table of contents HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015748979&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Ash, Avner Gross, Robert Fearless symmetry exposing the hidden patterns of numbers Nombres, Théorie des Number theory Symmetrie (DE-588)4058724-1 gnd Zahlentheorie (DE-588)4067277-3 gnd Darstellungstheorie (DE-588)4148816-7 gnd |
| subject_GND | (DE-588)4058724-1 (DE-588)4067277-3 (DE-588)4148816-7 |
| title | Fearless symmetry exposing the hidden patterns of numbers |
| title_auth | Fearless symmetry exposing the hidden patterns of numbers |
| title_exact_search | Fearless symmetry exposing the hidden patterns of numbers |
| title_exact_search_txtP | Fearless symmetry exposing the hidden patterns of numbers |
| title_full | Fearless symmetry exposing the hidden patterns of numbers Avner Ash and Robert Gross |
| title_fullStr | Fearless symmetry exposing the hidden patterns of numbers Avner Ash and Robert Gross |
| title_full_unstemmed | Fearless symmetry exposing the hidden patterns of numbers Avner Ash and Robert Gross |
| title_short | Fearless symmetry |
| title_sort | fearless symmetry exposing the hidden patterns of numbers |
| title_sub | exposing the hidden patterns of numbers |
| topic | Nombres, Théorie des Number theory Symmetrie (DE-588)4058724-1 gnd Zahlentheorie (DE-588)4067277-3 gnd Darstellungstheorie (DE-588)4148816-7 gnd |
| topic_facet | Nombres, Théorie des Number theory Symmetrie Zahlentheorie Darstellungstheorie |
| url | http://www.loc.gov/catdir/enhancements/fy0654/2005051471-b.html http://www.loc.gov/catdir/enhancements/fy0654/2005051471-d.html http://www.loc.gov/catdir/enhancements/fy0654/2005051471-t.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015748979&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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